A143641 -id:A143641 - cp's OEIS frontend

A118118 Composite numbers that always remain composite when a single decimal digit of the number is changed.

200, 204, 206, 208, 320, 322, 324, 325, 326, 328, 510, 512, 514, 515, 516, 518, 530, 532, 534, 535, 536, 538, 620, 622, 624, 625, 626, 628, 840, 842, 844, 845, 846, 848, 890, 892, 894, 895, 896, 898, 1070, 1072, 1074, 1075, 1076, 1078, 1130

Offset: 1

Adam Panagos (adam.panagos(AT)gmail.com), May 12 2006

The term "prime-proof" for this property is found on projecteuler.net (cf. link). The nontrivial subsequence A143641 is that of odd elements not ending in 5 (i.e. not ending in 0,2,4,5,6 or 8); it starts 212159,595631,872897,... - M. F. Hasler, Sep 04 2008

Also indices n such that A209252(n) is zero. - Ray G. Opao, Aug 01 2020

			a(1) = 200 is in the sequence because changing any digit of 200 (for example 300, 220, or 209) is still composite. The integer 100 is not in the sequence because it can be changed to 107 which is prime.

Paolo P. Lava, Table of n, a(n) for n = 1..10000
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 200
Project Euler, Problem 200. Find the 200th prime-proof sqube containing the contiguous sub-string "200" (2008)

Cf. A143641, A050249, A209252.

IsA118118:=function(n); D:=Intseq(n); return forall{ : k in [1..#D], j in [0..9] | j eq D[k] or not IsPrime(Seqint(S)) where S:=Insert(D, k, k, [j]) }; end function; [ n: n in [1..1200] | IsA118118(n) ]; // Klaus Brockhaus, Feb 28 2011

unprimeableQ[n_] := Block[{d = IntegerDigits@ n, t = {}}, Do[AppendTo[t, FromDigits@ ReplacePart[d, i -> #] & /@ DeleteCases[Range[0, 9], x_ /; x == d[[i]]]], {i, Length@ d}]; ! AnyTrue[Flatten@ t, PrimeQ]]; Select[Range@ 1200, unprimeableQ] (* Michael De Vlieger, Nov 09 2015, Version 10 *)

/* return 1 if no digit can be changed to make it prime; if d=1, print a prime if n is not prime-proof */ isA118118(n,d=0)={ forstep( k=n\10*10+1, n\10*10+9,2, isprime(k) || next; d && print("prime:",k); return); if( n%2==0 || n%5==0, /* even or ending in 5: no other digit can make it prime, except for the case where the last digit is prime and the first digit is the only other nonzero one */ return( !isprime(n%10) || 9 < n % 10^( log(n+.5)\log(10) ) || (d && print("prime:",n%10)) )); o=10; until( n < o*=10, k=n-o*(n\o%10); for( i=0,9, isprime(k) && return(d && print("prime:",k)); k+=o));1} \\ M. F. Hasler, Sep 04 2008

from sympy import isprime
def selfplusneighs(n):
    s = str(n); d = "0123456789"; L = len(s)
    yield from (int(s[:i]+c+s[i+1:]) for c in d for i in range(L))
def ok(n): return all(not isprime(k) for k in selfplusneighs(n))
print([k for k in range(1131) if ok(k)]) # Michael S. Branicky, Jun 19 2022

Edited by Charles R Greathouse IV, Aug 05 2010

A192545 Numbers such that all numbers are composite when replacing exactly one digit with another, except the leading digit with zero.

Original entry on oeis.org

200, 202, 204, 205, 206, 208, 320, 322, 324, 325, 326, 328, 510, 512, 514, 515, 516, 518, 530, 532, 534, 535, 536, 538, 620, 622, 624, 625, 626, 628, 840, 842, 844, 845, 846, 848, 890, 892, 894, 895, 896, 898, 1070, 1072, 1074, 1075, 1076, 1078, 1130, 1132

Offset: 1

Reinhard Zumkeller, Jul 05 2011

A048853(a(n)) = 0;
Intersection of this sequence and A000040 is A158124. - Evgeny Kapun, Dec 13 2016
If the last digit of an element is 0, 2, 4, 5, 6 or 8, then replacing it with 0, 2, 4, 5, 6 or 8 also yields an element. - David A. Corneth and corrected by Evgeny Kapun, Dec 13 2016

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Subsequences: A050249, A118118, A143641, A158124, A158125.

import Data.List (elemIndices)
a192545 n = a192545_list !! (n-1)
a192545_list = map (+ 1) $ elemIndices 0 $ map a048853 [1..]

Select[Range@ 1200, Function[w, Total@ Boole@ Flatten@ Map[Function[d, PrimeQ@ FromDigits@ ReplacePart[w, d -> #] & /@ If[d == 1, #, Prepend[#, 0]] &@ Range@ 9], Range@ Length@ w] == 0]@ IntegerDigits@ # &] (* Michael De Vlieger, Dec 13 2016 *)

A186694 Numbers ending in 1, 3, 7 or 9 such that changing any one decimal digit produces a composite number.

Original entry on oeis.org

212159, 294001, 505447, 584141, 595631, 604171, 872897, 971767, 1062599, 1203623, 1282529, 1293671, 1524181, 1566691, 1702357, 1830661, 2017963, 2474431, 2690201, 3085553, 3326489, 3716213, 3964169, 4103917, 4134953, 4173921, 4310617, 4376703

Offset: 1

Arkadiusz Wesolowski, Feb 25 2011

Union of A050249 and A143641.

This sequence is infinite because Terence Tao proved that sequence A050249 is infinite.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1500
Chris Caldwell, The Prime Glossary, Weakly prime
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 212159
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 17171...58369 (1000-digits)
Terence Tao, A remark on primality testing and decimal expansions, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.

Cf. A050249, A045572, A143641.

primeProof[n_] := Module[{d, e, isPP, num}, d=IntegerDigits[n]; isPP=True; Do[e=d; e[[i]]=j; num=FromDigits[e]; If[num != n && PrimeQ[num], isPP=False; Break[]], {i, Length[d]}, {j, 0, 9}]; isPP]; Select[Range[1, 1000000, 2], Mod[#, 5] > 0 && primeProof[#] &] (* T. D. Noe, Feb 26 2011 *)

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A118118 Composite numbers that always remain composite when a single decimal digit of the number is changed.

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A192545 Numbers such that all numbers are composite when replacing exactly one digit with another, except the leading digit with zero.

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A186694 Numbers ending in 1, 3, 7 or 9 such that changing any one decimal digit produces a composite number.

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Mathematica